Ricci Curvature and the Manifold Learning Problem

TL;DR

This paper introduces a method to estimate Ricci curvature of a submanifold from i.i.d. samples using diffusion semi-group theory, empirical processes, and local PCA, advancing manifold learning techniques.

Contribution

It presents a novel approach to estimate Ricci curvature directly from data samples, bridging geometric analysis and data-driven methods.

Findings

Successfully estimates Ricci curvature from finite samples

Integrates diffusion semi-group concepts with empirical processes

Provides theoretical guarantees for the estimation method

Abstract

Consider a sample of $n$ points taken i.i.d from a submanifold $\Sigma$ of Euclidean space. We show that there is a way to estimate the Ricci curvature of $\Sigma$ with respect to the induced metric from the sample. Our method is grounded in the notions of Carr\'e du Champ for diffusion semi-groups, the theory of Empirical processes and local Principal Component Analysis.